Questions tagged [groupoids]

A groupoid (in the sense of category theory) is a small category in which every morphism is an isomorphism. Groupoids arise throughout mathematics, e.g. in guise of fundamental groupoids in the theory of covering spaces, holonomy groupoids in the theory of foliations or Lie groupoids in mathematical physics. For groupoids in the sense of universal algebra, i.e., a set with a binary operation, please use the (magma) tag.

Filter by
Sorted by
Tagged with
0
votes
0answers
15 views

Inverse of a product in a groupoid

I know it is too trivial question to ask. Let $G$ be a groupoid. I am following the algebraic defition not the categorical one. $(x,y)$ is a composable pair $\iff$ $s(x)=r(y)$ which will imply $(y^{-1}...
0
votes
0answers
11 views

product in Groupoids

Let $G$ be a groupoid and let $G^{(2)}$ , $G^{(0)}$ denotes its composable pairs and unit space respectively. We have $(x,y) \in G^{(2)}\Leftrightarrow s(x)=r(y)$ where $s$ and $r$ are source and ...
3
votes
1answer
71 views

Quotient by a topological groupoid.

Let $G$ be a group acting on a topological space $X$, then the quotient map $X \to X/G$ is open. I want to ask, whether this fact generalizes to orbit spaces of groupoids. More precisely: Let $G$ be a ...
0
votes
0answers
5 views

How are two embeddings of orbifolds are related?

The following Lemma is used to prove that smooth embedding of orbifold charts give rise to injective homomorphism in groups acting on respective manifolds, how do I prove following: Given 2 embeddings ...
1
vote
2answers
46 views

$G$-set are groupoids of a fibred category

Take the category of all G-sets for different groups $G$. Each $G$-set is the groupoid $G \times \Omega \to \Omega$ (the first projection is the source and the action map is the target). I am not ...
2
votes
2answers
88 views

Why can't the 3-type of $S^2$ be delooped?

In 2.2 of this paper, the argument runs through the fact that the 3-type of $S^2$ cannot be delooped. Why not? I understand that this is probably a fairly basic fact of homotopy theory (hence neither ...
2
votes
0answers
120 views

Existence of universal covering groupoids

I am not very experienced in this topic, so please excuse me if there are any stupid mistakes. Let $G$ be a groupoid. In “Calculus of fractions and Homotopy Theory” by Gabriel and Zisman it is shown, ...
1
vote
0answers
33 views

Existence of horizontal inverses in 2-grupoids

In the paper "An Invitation to Higher Gauge Theory" by Baez and Huerta 2-grupoid is defined as a 2-category in which every morphism is invertible and every 2-morphism has a vertical inverse. It is ...
0
votes
0answers
33 views

General linear groupoid $GL(E)$ is not proper and $O(E)$ is proper

I'm studying groupoids and I'm having trouble with the following exercise: Example 1: If $E$ is a vector bundle over $M$, there is an associated general linear groupoid, denoted by $GL(E)$, which ...
2
votes
0answers
29 views

Number of fixed point free permutations by groupoid cardinals

It is well known that the number $D_n$ of derangements (fixed point free permutations) on a set of $n$ elements is exactly $\left[\dfrac{n!}e\right]$, the closest integer number to $\dfrac{n!}e$. ...
8
votes
1answer
155 views

$\pi_1(S^1, 1)$ via the fundamental groupoid

I'm currently reading Ronnie Brown's Topology and Groupoids and am stuck on a small detail of his computation of the fundamental group of the circle (in particular his computation of the group's ...
2
votes
1answer
37 views

Functor from $BG \to \text{Gpd}$ induces functors for each $g \in G$.

I'm reading Groupoids and Stuff. Page 7. Definition 1.3.1. A group action of a a finite group $G$ on a groupoid $X$ is a functor $A : B G \to \text{Gpd}$ such that $A(I) = X$ where $I$ is the unique ...
1
vote
1answer
38 views

Groupoids all of whose subcategories are themselves groupoids

It is known that every submonoid of a group $G$ is a subgroup if and only if $G$ is a periodic group, i.e. all of its elements have finite order. The following question is a generalization of the ...
0
votes
1answer
25 views

How to realize the map $\eta$ globally?

I have a given map $\Phi: \mathcal{G}\longrightarrow \mathcal{H}$ between two groupoids such that $\Phi_g: \mathcal{G}_x\longrightarrow \mathcal{G}_y$ is a functor between the groupoids $\mathcal{G}_x$...
7
votes
1answer
112 views

Riehl's Category Theory in Context - Exercise 1.5.vii without Axiom of Choice

From Emily Riehl, Category theory in context: Exercise 1.5.vii. Let $\mathbf{\mathsf G}$ be a connected groupoid and let $G$ be the group of automorphisms at any of its objects. The inclusion $\...
9
votes
0answers
58 views

Exemples of applications of “groupoidification” to linear algebra

I just read Baez's very nice blog notes about groupoidification, and around the beginning, he states : "From all this, you should begin to vaguely see that starting from any sort of incidence ...
1
vote
0answers
31 views

Is the 2-Category of Groupoids Locally Presentable?

I am wondering if the 2-Category of groupoids is Locally Presentable? Locally presentable means the category is accessible and co-complete. Edit: It has been pointed out that the category of ...
2
votes
0answers
61 views

Why are categories not called monoids (and why are monoids not called… mons?) [closed]

If groupoids are "indexed groups", wouldn't that same naming scheme imply that categories should be called "monoidoids", or more sensibly, why aren't categories called "monoids" and monoids called... "...
7
votes
1answer
60 views

What's the correct notion of equivalence in a double category?

What's the appropriate notion of equivalence between two objects in a double category? At first I thought the answer was just an equivalence in one of the associated $2$-categories, but then I ...
2
votes
0answers
40 views

Equivalence of Lie groupoids $\phi: H \rightarrow G$ induces an equivalence of categories $\phi^*: G\text{-spaces} \rightarrow H\text{-spaces}$.

In Orbifolds as Groupoids there is the notion of an equivalence $\phi: H \rightarrow G$ between Lie groupoids (2.4) and of $G$-spaces (5.1). Given a smooth functor $\phi: H \rightarrow G$ we can ...
2
votes
1answer
56 views

Equivalence between the Dwyer-Kan loop groupoid and the fundamental groupoid

Let $X$ be a homotopy 1-type (a space with vanishing homotopy groups above degree one). It is a classical fact that $X$ can be recovered completely from its fundamental groupoid. On the the other ...
2
votes
0answers
36 views

Inversion on an internal groupoid

Let $\mathcal{C}$ be a category with pullbacks, with $\mathscr{G}=({\bf Ob}_\mathscr{G},{\bf Hom}_\mathscr{G},cod,dom,{\bf 1}, \circ_{\mathscr{G}},-^{-1})$ an internal groupoid in $\mathcal{C}$. I'm ...
1
vote
0answers
71 views

Faithful representation of $C_c(X \times X)$.

Let $X$ be a smooth manifold. $X \times X$ is product manifold. $\mu$ is a Borel measure on $X$. There are two aims (i) Associate a $C^*$ norm to $C_c(X\times X)$, making it a $C^*$ algebra. (ii)...
1
vote
0answers
38 views

Groupoid Theory I, Higson's notes

This is Example 7.22, page 100. We first consider $G=TM$ a smooth groupoid with base space $M$. The source and range maps are the same projection maps $s,r:TM \rightarrow M, X_m \mapsto m$. Now I ...
2
votes
1answer
77 views

Milnor construction and deloopings

To construct a classifying space (and universal bundle) of a topological group $G$ one can use the well-known Milnor construction based on the infinite join of $G$. On the other hand one can (at ...
0
votes
0answers
147 views

What is a topological groupoid?

I'm reading section 2.7 (Fundamental Group of the Circle) of the book Algebraic Topology by Tom Dieck. The section mentions the term "topological groupoid" but I cannot find the definition in previous ...
0
votes
1answer
105 views

Is the 2-category of groupoids a topos?

I have no justification for this, but I am wondering if the 2-category of groupoids is a topos.
0
votes
1answer
61 views

Object of a Category $C$ acts as Functor

I have a question about a notation used in following paper: https://etale.site/writing/stax-seminar-talk.pdf (see page 4): We take a category $C$ and consider a pair $(X_0,X_1)$ of two objects $X_1, ...
2
votes
0answers
103 views

Categories Fibered in Groupoids and Yoneda

My question refers to an article of Aaron Mazel-Gee about fibered categories in grupoids where the author introduces in a quite strange way a new category $\{X_0/X_1\}$ providing a functor $p: \{X_0/...
1
vote
1answer
68 views

Space of Riemannian metrics as a topological groupoid

I'm reading these notes on groupoids and I'm struggling with example 1.4. I recall the relevant definitions below. Definition: A groupoid $\mathcal{G}$ is a small category in which every arrow is ...
1
vote
2answers
107 views

Fibered Categories in Groupoids

I'm reading an article of Aaron Mazel-Gee about Fibered categories in grupoids and there is an example which I don't understand. Here the full article: https://etale.site/writing/stax-seminar-talk.pdf ...
2
votes
1answer
61 views

Can nonisomorphic groupoids have homotopy equivalent classifying spaces?

We know that two discrete groups having the same classifying space up to homotopy are isomorphic. One can just take fundamental groups and conclude. The situation with topological groups is subtler. ...
4
votes
1answer
72 views

Right (bi)adjoint of the inclusion of $\mathbf{Grpd}$ in $\mathbf{Cat}$

Let $\mathbf{Grpd}$ and $\mathbf{Cat}$ be respectively the 2-categories of small groupoids and of small categories. At the 1-categorical level, the inclusion $\mathbf{Grpd}\rightarrow\mathbf{Cat}$ has ...
5
votes
1answer
402 views

what is an ∞-group?

I was reading on nLab and I found the term infinity group. The definition is awfully abstract: An ∞-group is a group object in ∞Grpd. Equivalently (by the delooping hypothesis) it is a pointed ...
3
votes
1answer
65 views

Is every topological groupoid equivalent to a disjoint union of topological groups?

It's a fact that any groupoid is equivalent to a disjoint union of (deloopings of) groups. See, e.g. Proposition 4.3 of https://ncatlab.org/nlab/show/groupoid#PropertiesEquivalencesOfGroupoids. Does ...
2
votes
1answer
155 views

The Whitehead product and $\pi_{\leq 3} S^2$

Why does "the non-vanishing of the Whitehead bracket" imply that the fundamental 3-groupoid $\pi_{\leq 3} S^2$ of the two-shere cannot be strictified (as claimed here)?
1
vote
0answers
30 views

The $\mathbb{C}$ vector space monad on the 2-Category of Groupoids

In this post, I am asking about the existence of something called the "Vector Space Monad" on the 2-Category of groupoids (Grpd). In a comment, it was pointed out that the monad should exist due to ...
1
vote
0answers
65 views

Is there a simplicial set classifying subobjects of groupoids?

A $1$-groupoid can be thought of as a Kan complex in the usual way. Is there a simplicial set $\Omega$ such that the contravariant functors $\text{Sub}_{\mathbf{Gpd}}(-)$ and $\text{Hom}_{\mathbf{sSet}...
-1
votes
1answer
32 views

Can we see FdHlb as a 2Category of groupoids?

Can we see a finite dimensional Hilbert space, $H$ as a groupoid if we include the unitary endomorphisms of $H$? It would be like a category with a single object and just isos. If so, can we take a ...
0
votes
1answer
111 views

Equivalence of Categories between the Fundamental Group and Groupoid

Let $X$ be a path connected space, and let $x \in X$. Then we have that $\pi_1 (X,x)$ is a full subcategory of $\Pi(X)$. So, the inclusion functor $J: \pi_1(X,x) \to \Pi(X)$ is an equivalence of ...
0
votes
1answer
50 views

Definition of $\pi_0 p^{-1}(u)$

In Ronnie Brown's Topology and groupoid, pg 263, 7.2.1 If $p:E\to B$ is a fibration of groupoids, there is an assignment, $$b_\#:\pi_0 p^{-1}[u]\to\pi_0 p^{-1}[v]$$ I did not see anywhere where ...
3
votes
2answers
102 views

“Eilenberg-MacLane property” for the classifying space of a groupoid

Given a groupoid $G$, its classifying space is defined as the standard geometric realisation of the nerve. My question is: since the classifying space of a group is the only space up to homotopy that ...
2
votes
1answer
65 views

All sheaves in a Grothendieck topos are generated by subobjects of powers of a suitable sheaf

Consider a topos $\mathcal E$. Butz and Moerdijk, in Representing topoi by topological groupoids, par. 2, say that one can find an object $S\in \mathcal E$ such that the subobjects of its powers (i.e.,...
5
votes
1answer
181 views

Morita-equivalence of groupoids and classifying topoi: correct definition

The first comment to this post points out that, given two (topological or localic) groupoids, they can be non-Morita-equivalent evenif their classifying topoi (topoi of equivariant sheaves) are ...
1
vote
0answers
60 views

Understanding the monadicity of groupoids over splittings

In the paper The shift functor and the comprehensive factorization for internal groupoids by Bourn, the author proves that for a fixed finitely complete category, the category of internal groupoids is ...
4
votes
3answers
198 views

Higher homotopy groups in terms of the fundamental groupoid

Let $X$ be a topological space. Then we can construct the following structure. Let an $n$-morphism be a map $I^n\to X$. We can view $n+1$ morphisms exactly as homotopies between $n$-morphisms. Let $f,...
11
votes
0answers
217 views

The étale topos of a scheme is the classifying topos of…?

By a theorem of Joyal and Tierney, every Grothendieck topos is the classifying topos of a localic groupoid. It has been proved (e.g. C. Butz. and I. Moerdijk. Representing topoi by topological ...
2
votes
1answer
93 views

Are there examples of unital and nuclear $C^*$-algebras satisfying the UCT that are not groupoid algebras of an amenable etale groupoid?

Jean Louis Tu showed that the (maximal) groupoid $C^*$-algebra of a groupoid satisfying the Haagerup property (which includes all amenable groupoids) will satisfy the UCT. I am curious if there are ...
6
votes
0answers
128 views

Foliations and groupoids in algebraic geometry

I am currently studying the theory of foliations and groupoids from a differentiable viewpoint, in particular Haefliger spaces. [See Segal, Classifying spaces related to foliations, and Moerdijk, ...
0
votes
0answers
41 views

Subsets of groups containing identity and inverses

Let $G$ be a (finite) group, containing a subset $H$. We suppose that $H$ contains the identity and that it is closed under taking inverses. What are some of the algebraic properties of subgroups ...